How to move a whole mechanical rig without selecting all the individual parts?
Rigid Body objects are all automatically assigned to a specific Rigid Body group. To select them all simply select one and then use Shift+G to select group. See https://docs.blender.org/manual/en/dev/editors/3dview/object/selecting/tools.html#bpy-ops-object-select-grouped

Once selected simply use the usual manipulation tools (eg grab, rotate) to move them.

You can also create your own groups to allow you to organise sub-sections of your objects and select them in the same way.